Minimum (2, r)-Metrics and Integer Multiflows

Let H=(T, U)be a connected graph. A T-partitionof a set V⊇Tis a partition of Vinto subsets, each containing exactly one element of T. We start with the following problem (*): given a multigraph G=(V, E)with V⊇T,find a T-partition Π of Vthat minimizes the sum of products d(s, t)n(s, t)over all s,t∈T.Here d(s, t)is the distance from sto tin Hand n(s, t)is the number of edges of Gbetween the sets in Π that contain sand t.When the graph His complete, (*) turns into the minimum multiway cut problem, which is known to be NP-hard even if |T|=3.On the other hand, when His the complete bipartite graph K 2 , r with parts of 2 and r=|T|−2nodes, (*) is specialized to be the minimum (2, r)-metric problem, which can be solved in polynomial time. We prove that the multicommodity flow problem dual of the minimum (2, r)-metric problem has an integer optimal solution whenever Gis inner Eulerian(i.e. the degree of each node in V−Tis even), and such a solution can be found in polynomial time. Another nice property of K 2 , r is that, independently of G,the optimum objective value in (*) is the same as that in its factional relaxation. We call a graph Hwith a similar property minimizableand give a description of the minimizable graphs in polyhedral terms. Finally, we show that every tree is minimizazble.